Generalizations of Hardy's integral inequalities (Q1587810)

The authors study a question whether the constant factors involved in the following Hardy-type inequalities are the best possible: \[ \begin{aligned} \int^b_a \Big( x^{-1} \int^x_a f(t) dt\Big)^p dx &\leq q^p \Big[ 1-\Big(\frac{a}{b}\Big)^{1/q}\Big]^p \int^b_a f^p(t) dt,\\ \int^{\infty}_a \Big( x^{-1} \int^x_a f(t) dt\Big)^p dx &\leq q^p \int^{\infty}_a [1-\theta_p(t)] f^p(t) dt,\\ \int^b_0\Big(x^{-1} \int^x_0 f(t)dt\Big)^p dx &\leq q^p\int^b_0 \Big[ 1- \Big(\frac{t}{b}\Big)^{1/q}\Big] f^p(t) dt. \end{aligned} \] Here \(0<a<b<\infty\), \(p\in (1,\infty)\), \(q=p /(p-1)\) and \(\theta_p\) is a convenient function such that \(0 < \theta_p (t) < 1\) for all \(t\in (a,\infty)\). To answer this question, the authors establish some generalizations of the inequalities mentioned above.